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Improved distances and ages for stars common to TGAS and RAVEMcMillan, Paul J.; Kordopatis, Georges; Kunder, Andrea; Binney, James; Wojno, Jennifer;Zwitter, Tomaz; Steinmetz, Matthias; Bland-Hawthorn, Joss; Gibson, Brad K.; Gilmore,GerardPublished in:Monthly Notices of the Royal Astronomical Society

DOI:10.1093/mnras/sty990

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Citation for published version (APA):McMillan, P. J., Kordopatis, G., Kunder, A., Binney, J., Wojno, J., Zwitter, T., Steinmetz, M., Bland-Hawthorn, J., Gibson, B. K., Gilmore, G., Grebel, E. K., Helmi, A., Munari, U., Navarro, J. F., Parker, Q. A.,Seabroke, G., Watson, F., & Wyse, R. F. G. (2018). Improved distances and ages for stars common toTGAS and RAVE. Monthly Notices of the Royal Astronomical Society, 477(4), 5279-5300.https://doi.org/10.1093/mnras/sty990

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https://doi.org/10.1093/mnras/sty990https://research.rug.nl/en/publications/improved-distances-and-ages-for-stars-common-to-tgas-and-rave(f4f71ef9-47fe-4272-9929-4496375d6ac6).htmlhttps://doi.org/10.1093/mnras/sty990

MNRAS 477, 5279–5300 (2018) doi:10.1093/mnras/sty990Advance Access publication 2018 April 21

Improved distances and ages for stars common to TGAS and RAVE

Paul J. McMillan,1‹ Georges Kordopatis,2 Andrea Kunder,3 James Binney,4

Jennifer Wojno,3,5 Tomaž Zwitter,6 Matthias Steinmetz,3 Joss Bland-Hawthorn,7

Brad K. Gibson,8 Gerard Gilmore,9 Eva K. Grebel,10 Amina Helmi,11 Ulisse Munari,12

Julio F. Navarro,13 Quentin A. Parker,14,15 George Seabroke,16 Fred Watson17 andRosemary F. G. Wyse51Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, SE-22100 Lund, Sweden2Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Parc Valrose, F-06108 Nice, France3Leibniz-Institut für Astrophysik Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany4Rudolf Peierls Centre for Theoretical Physics, Keble Road, Oxford OX1 3NP, UK5Department of Physics and Astronomy, Johns Hopkins University, 3400 N. Charles St, Baltimore, MD 21218, USA6Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia7Sydney Institute for Astronomy, School of Physics A28, University of Sydney, NSW 2006, Australia8E.A. Milne Centre for Astrophysics, University of Hull, Hull HU6 7RX, UK9Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK10Astronomisches Rechen-Institut, Zentrum für Astronomie der Universität Heidelberg, Mönchhofstr. 12–14, D-69120 Heidelberg, Germany11Kapteyn Astronomical Institute, University of Groningen, PO Box 800, NL-9700 AV Groningen, the Netherlands12INAF Astronomical Observatory of Padova, I-36012 Asiago (VI), Italy13Senior CIfAR Fellow, Department of Physics and Astronomy, University of Victoria, Victoria BC V8P 5C2, Canada14Department of Physics, The University of Hong Kong, Hong Kong SAR, China15The University of Hong Kong, Laboratory for Space Research, Hong Kong SAR, China16Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK17Australian Astronomical Observatory, PO Box 915, North Ryde, NSW 1670, Australia

Accepted 2018 April 18. Received 2018 April 17; in original form 2017 July 14

ABSTRACTWe combine parallaxes from the first Gaia data release with the spectrophotometric distanceestimation framework for stars in the fifth RAVE survey data release. The combined distanceestimates are more accurate than either determination in isolation – uncertainties are on averagetwo times smaller than for RAVE-only distances (three times smaller for dwarfs), and 1.4 timessmaller than TGAS parallax uncertainties (two times smaller for giants). We are also able tocompare the estimates from spectrophotometry to those from Gaia, and use this to assess thereliability of both catalogues and improve our distance estimates. We find that the distances tothe lowest log g stars are, on average, overestimated and caution that they may not be reliable.We also find that it is likely that the Gaia random uncertainties are smaller than the reportedvalues. As a by-product we derive ages for the RAVE stars, many with relative uncertaintiesless than 20 per cent. These results for 219 566 RAVE sources have been made publiclyavailable, and we encourage their use for studies that combine the radial velocities providedby RAVE with the proper motions provided by Gaia. A sample that we believe to be reliablecan be found by taking only the stars with the flag notification ‘flag any=0’.Key words: methods: statistical – Galaxy: fundamental parameters – Galaxy: kinematics anddynamics – Galaxy: structure.

1 IN T RO D U C T I O N

ESA’s Gaia mission (Gaia Collaboration 2016a) is an enormousproject that is revolutionizing Milky Way astronomy. Gaia will

� E-mail: [email protected]

provide a wide range of data about the stars of the Milky Way, in-cluding photometry and spectroscopy. However it is the astrometry– and in particular the parallaxes – from Gaia that are the cause ofthe most excitement. It is very difficult to determine the distances tostars, and not knowing the distance to a star means that one knowsneither where it is nor how fast it is moving, even if the propermotion of the star is known.

C© 2018 The Author(s)Published by Oxford University Press on behalf of the Royal Astronomical Society

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mailto:[email protected]

5280 P. J. McMillan et al.

The RAVE survey (Radial Velocity Experiment; Steinmetz et al.2006) is a spectroscopic survey that took spectra for ∼500 000stars. From these one could determine for each star its line-of-sight velocity and the structural parameters, such as its effectivetemperature (Teff), surface gravity (log g), and metallicity ([M/H]).These can be used to derive the distances to stars, and since RAVE’sfourth data release (Kordopatis et al. 2013) these have been providedby the Bayesian method that was introduced by Burnett & Binney(2010), and extended by Binney et al. (2014). Bayesian methodshad previously been used for distance estimation in astrophysicsfor small numbers of stars of specific classes (Barnes et al. 2003;Thorstensen 2003), and the Burnett & Binney method is similar toan approach that had previously been used to determine the agesof stars (Pont & Eyer 2004; Jørgensen & Lindegren 2005). Closelyrelated approaches have since been used by numerous studies (e.g.Serenelli et al. 2013; Schönrich & Bergemann 2014; Santiago et al.2016; Wang et al. 2016; Mints & Hekker 2017; Queiroz et al.2017; Schneider et al. 2017). The method produces a probabilitydensity function (pdf) for the distance, and these pdfs were testedby, amongst other things, comparison of some of the correspondingparallax estimates to the parallaxes found by Gaia’s predecessorHipparcos (Perryman et al. 1997; van Leeuwen 2007). RAVE’smost recent data release was the fifth in the series (henceforth DR5),and included distance estimates found using this method (Kunderet al. 2017). The RAVE sample appears to be kinematically andchemically unbiased (Wojno et al. 2017).

Gaia’s first data release (Gaia DR1; Gaia Collaboration 2016b;Lindegren et al. 2016) includes parallaxes and proper motions for∼ 2000 000 sources. These were available earlier than full astrom-etry for the other ∼1 billion sources observed by Gaia, because thesources were observed more than 20 yr earlier by the Hipparcosmission, and their positions at that epoch (and proper motions) ap-pear in either the Hipparcos catalogue or the, less precise, Tycho-2catalogue (Høg et al. 2000), which used data from the Hipparcossatellite’s star mapper. This means that the proper motions of thestars can be derived using this very long time baseline, which breaksdegeneracies between proper motion and parallax that made the de-termination of these parameters for the other sources impossible.The resulting catalogue is known as the Tycho-Gaia Astrometricsolution (TGAS; Michalik, Lindegren & Hobbs 2015).

Since RAVE and TGAS use fundamentally different methods forderiving the distances to stars, it is inevitable that these have dif-ferent precisions for different types of stars. The Burnett & Binney(2010) method relies, fundamentally, on comparing the observedmagnitude to the expected luminosity. The uncertainty in distancemodulus, which is roughly equivalent to a relative distance uncer-tainty, is therefore approximately independent of the distance to thestar. The parallax uncertainty from TGAS, on the other hand, isindependent of the parallax value, so the relative precision declineswith distance – large distances correspond to small parallaxes, andtherefore large relative uncertainties.

In Fig. 1 we show the quoted parallax uncertainty from bothTGAS and DR5 for the sources common to both catalogues. Inthe case of TGAS we use the quoted statistical uncertainties (seeSection 6 for further discussion). We also divide this into the un-certainty for giant stars (DR5 log g < 3.5) and dwarfs (DR5 log g≥ 3.5). We see that for TGAS this distinction is immaterial, whileit makes an enormous difference for DR5. The DR5 parallax esti-mates tend to be less precise than the TGAS ones for dwarfs (whichtend to be nearby because the survey is magnitude limited), but asprecise, or more, for the more luminous giants, especially the moredistant ones.

Figure 1. Histograms of the quoted random parallax uncertainties (σ� )from TGAS and those from RAVE DR5 for stars common to the two cat-alogues. We show histograms of the uncertainties for all stars (solid), andseparately for giants (log gDR5 < 3.5) and dwarfs (log gDR5 ≥ 3.5). They-axis gives the number of stars per bin, and there are 40 bins in total inboth cases. The cut-off at 1 mas for the TGAS parallaxes is due to a filterapplied by the Gaia consortium to their DR1. For RAVE sources we makethe standard cuts to the catalogue described in Kunder et al. (2017). TGASparallaxes are more precise than RAVE’s for dwarfs, but not necessarily forgiants.

It is worth noting that TGAS provides only parallax measure-ments, not distance estimates and, as discussed by numerous au-thors at various points over the last century, the relationship betweenone and the other is non-trivial when one takes the uncertainties intoaccount (e.g. Strömberg 1927; Lutz & Kelker 1973; Luri & Are-nou 1997; Bailer-Jones 2015). Astraatmadja & Bailer-Jones (2016)looked at how the distances derived from TGAS parallaxes dependon the prior probability distribution used for the density of stars, butdid not use any information about a star other than its parallax.

For this reason, and because TGAS parallaxes have large relativeerrors for distant stars, when studying the dynamics of the MilkyWay using stars common to RAVE and TGAS, it has been seenas advantageous to use distances from DR5 rather than those fromTGAS parallaxes (e.g. Hunt, Bovy & Carlberg 2016; Helmi et al.2017). It is therefore important to improve these distance estimatesand to check whether there are any systematic errors associated withthe DR5 distance estimates.

Kunder et al. (2017) discuss the new efforts in RAVE DR5 toreconsider the parameters of the observed stars. They provided newTeff values derived from the Infrared Flux Method (IRFM; Black-well, Shallis & Selby 1979) using an updated version of the im-plementation described by Casagrande et al. (2010). Also providedin a separate data-table were new values of log g following a re-calibration for red giants from the Valentini et al. (2017) study of72 stars with log g values derived from asteroseismology of starsby the K2 mission (Howell et al. 2014). These were not used to de-rive distances in the main DR5 catalogue, and we now explore howusing these new data products can improve our distance estimates.

In this study, we compare parallax estimates from TGAS andRAVE to learn about the flaws in both catalogues. We then includethe TGAS parallaxes in the RAVE distance estimation, to derivemore precise distance estimates than are possible with either set ofdata in isolation.

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Distances and ages for RAVE-TGAS stars 5281

It is also possible to derive ages for stars from the same efforts,indeed the use of Bayesian methods to derive distances was pre-ceded by studies using them to determine ages (Pont & Eyer 2004;Jørgensen & Lindegren 2005). RAVE DR4 included the age esti-mates derived alongside the distances, but these were recognized asonly being indicative (Kordopatis et al. 2013). In this study we showthe substantial improvement that is possible using TGAS parallaxesand a more relaxed prior.

In Section 2 we describe the method used to derive distances. InSection 3 we compare results from DR5 to those from TGAS, whichmotivates us to look at improving our parallax estimates using otherRAVE data products in Section 4. In Section 5 we explore the effectof varying our prior. In Section 6 we look at what we can learn aboutTGAS by comparison with these new parallax estimates. Finally,Sections 7, 8, and 9 demonstrate the improvements made possibleby using the TGAS parallaxes as input to the Bayesian scheme.

2 BAY ESIAN ESTIMATION

Since RAVE DR4, distances to the stars in the RAVE survey havebeen determined using the Bayesian method developed by Burnett& Binney (2010). This takes as its input the stellar parameters Teff,log g, and [M/H] determined from the RAVE spectra, and J, H, andKs magnitudes from 2MASS (Skrutskie et al. 2006). This methodwas extended by Binney et al. (2014) to include dust extinction inthe modelling, and introduce an improvement in the description ofthe distance to the stars by providing multi-Gaussian fits to the fullpdf in distance modulus.1

In this paper we extend this method, principally by including theparallaxes found by TGAS as input, but also by adding AllWISEW1 and W2 mid-infrared photometry (Cutri & et al. 2013). We willexplore improvements made possible by using IRFM Teff valuesgiven in RAVE DR5, rather than Teff derived from the spectra. Weexpect that the IRFM values can be more precise than those fromthe RAVE spectra, which only span a narrow range in wavelength(8410–8795 Å).

Because the original intention of this pipeline was to estimatedistances, we often refer to it as the ‘distance pipeline’. In practicewe are now often as interested in its other outputs as we are inthe distance estimates. The pipeline applies the simple Bayesianstatement

P (model|data) = P (data|model)P (model)P (data)

, (1)

where in our case ‘data’ refer to the inputs described above (andshown in Table 1) for a single star, and ‘model’ comprises a starof specified initial mass M, age τ , metallicity [M/H], and locationrelative to the Sun (where Galactic coordinates l and b are treatedas known and distance s is unknown), observed through a spec-ified line-of-sight extinction, which we parametrize by extinctionin the V-band, AV. The likelihood P(data|model) is determined as-suming uncorrelated Gaussian uncertainties on all inputs, and usingisochrones to find the values of the stellar parameters and absolutemagnitudes of the model star. The isochrones that we use are fromthe PARSEC v1.1 set (Bressan et al. 2012), and the metallicitiesof the isochrones used are given in Table 2. P(model) is our priorwhich we discuss below, and P(data) is a normalization constant

1While the distance estimates always use 2MASS (and, in this study, All-WISE) photometry, we will refer to them as ‘RAVE-only’ at various pointsin this paper, to distinguish them from those found using TGAS parallaxesas input too.

Table 1. Data used to derive the distances to our stars, and their source.

Data Symbol Notes

Effectivetemperature

Teff RAVE DR5 – either from spectrum(DR5) or IRFM

Surface gravity log g RAVE DR5Metallicity [M/H] RAVE DR5J-band magnitude J 2MASSH-band magnitude H 2MASSKs-bandmagnitude

Ks 2MASS

W1-bandmagnitude

W1 AllWISE – not used for DR5 distances

W2-bandmagnitude

W2 AllWISE – not used for DR5 distances

Parallax � TGAS Gaia DR1 – not used for DR5 distancesor in comparisons

Table 2. Metallicities of isochrones used, taking Z� = 0.0152 and applyingscaled solar composition, with Y = 0.2485 + 1.78Z. Note that the minimummetallicity is [M/H] = −2.2, significantly lower than for the Binney et al.(2014) distance estimates where the minimum metallicity used was −0.9,which caused a distance underestimation for the more metal-poor stars(Anguiano et al. 2015).

Z Y [M/H]

0.00010 0.249 −2.2070.00020 0.249 −1.9060.00040 0.249 −1.6040.00071 0.250 −1.3550.00112 0.250 −1.1560.00200 0.252 −0.9030.00320 0.254 −0.6970.00400 0.256 −0.5980.00562 0.259 −0.4480.00800 0.263 −0.2910.01000 0.266 −0.1910.01120 0.268 −0.1390.01300 0.272 −0.0720.01600 0.277 0.0240.02000 0.284 0.1270.02500 0.293 0.2330.03550 0.312 0.4040.04000 0.320 0.4650.04470 0.328 0.5220.05000 0.338 0.5810.06000 0.355 0.680

which we can ignore. The assumption of uncorrelated Gaussian er-rors on the stellar parameters is one which is imperfect (see e.g.Schönrich & Bergemann 2014; Schneider et al. 2017), but it is thebest approximation that we have available for RAVE.

Putting this in a more mathematical form and defining the notationfor a single Gaussian distribution

G(x, μ, σ ) = 1√2πσ 2

exp

((x − μ)2

2σ 2

), (2)

we have

P (M, τ, [M/H], s, AV | data) ∝ P (M, τ, [M/H], s, AV |l, b)×

∏i

G(OTi (M, τ, [M/H], s, AV ), Oi, σi), (3)

where the prior P (M, τ, [M/H], s, AV |l, b) is described in Sec-tion 2.1, and the inputs Oi, σ i are those given in Table 1 (the cases



where any of these inputs are unavailable or not used can be treatedas the case where σ i → ∞). The theoretical values of these quan-tities – OTi (M, τ, [M/H], s, AV ) – are found using the isochronesand the relations between extinctions in different bands given inSection 2.1.

Once we have calculated the pdfs P(model|data) for the stars wecan characterize them however we wish. In practice, we characterizethem by the expectation values and standard deviation (i.e. estimatesand their uncertainties) for all parameters, found by marginalizingover all other parameters.

For distance we find several characterizations of the pdf: expec-tation values and standard deviation for the distance itself (s), fordistance modulus (μ), and for parallax � . The characterization interms of parallax is vital for comparison with TGAS parallaxes.

In addition we provide multi-Gaussian fits to the pdfs in distancemodulus because a number of the pdfs are multimodal, typically be-cause it is unclear from the data whether a star is a main-sequencestar or a (sub-)giant. Therefore, a single expectation value and stan-dard deviation is a poor description of the pdf. The multi-Gaussianfits to the pdfs in μ provide a compact representation of the pdf,and following Binney et al. (2014) we write them as

P (μ) =NGau∑k=1

fkG(μ, μ̂k, σk), (4)

where the number of components NGau, the means μ̂k , weights fk,and dispersions σ k are determined by the pipeline.

To determine whether a distance pdf is well represented by agiven multi-Gaussian representation in μ we take bins in distancemodulus of width wi = 0.2 mag, which contain a fraction pi of thetotal probability taken from the computed pdf and a fraction Pifrom the Gaussian representation, and compute the goodness-of-fitstatistic

F =∑

i

(pi

wi− Pi

wi

)2σ̃wi, (5)

where the weighted dispersion

σ̃ 2 ≡∑

k=1,NGaufkσ

2k (6)

is a measure of the overall width of the pdf. Our strategy is torepresent the pdf with as few Gaussian components as possible, butif the value of F is greater than a threshold value (Ft= 0.04), or thestandard deviation associated with the model differs by more than20 per cent from that of the complete pdf, then we conclude that therepresentation is not adequate, and add another Gaussian componentto the representation (to a maximum of three components, whichwe have found is almost always enough). We fit the multi-Gaussianrepresentation to the histogram using the Levenberg–Marquandtalgorithm (e.g. Press, Flannery & Teukolsky 1986), which we applymultiple times with different starting points estimated from themodes of the distribution. In this way we can take the best result andtherefore avoid getting caught in local minima. The relatively broadbins mean that we only use more than one Gaussian component ifthe pdf is significantly multimodal, though this comes at the cost ofreducing the accuracy of the fit when a peak is narrow.

These multi-Gaussian fits were particularly important in previ-ous RAVE data releases. In DR5 we found that a single Gaussiancomponent proved adequate for only 45 per cent of the stars, whilearound 51 per cent are fitted with two Gaussians, and only 4 per centrequire a third component. In Section 7 we show that the additionof TGAS parallaxes substantially reduces the number of stars forwhich more than one Gaussian is required.

The value of F is provided in the data base as FitQuality Gauss,and we also include a flag (denoted Fit Flag Gauss) which is non-zero if the standard deviation of the final fitted model differs bymore than 20 per cent from that of the computed pdf. Typically,the problems flagged are rather minor (as shown in fig. 3 of Binneyet al. 2014).

The uncertainties of the RAVE stellar parameters are assumed tobe the quadratic sum of the quoted internal uncertainties and theexternal uncertainties (table 4 of DR5). The external uncertaintiesare those calculated from stars with an SNR>40, except in the caseof the IRFM temperatures for which a single uncertainty serves forstars of every SNR since the IRFM temperatures are not extractedfrom the spectra. We discard all observations with a signal-to-noiseratio less than 10, or where the RAVE spectral pipeline returns aquality flag (AlgoConv) of ‘1’, because the quoted parameters forthese observations are regarded as unreliable.

For the 2MASS and AllWISE photometry we use the quoted un-certainties. We discard the AllWISE magnitudes if they are brighterthan the expected saturation limit in each band, which we take to beW1, sat= 8.1 mag, and W2, sat = 6.7 mag (following Cutri et al. 2012).

When using the TGAS parallaxes, we consider only the quotedstatistical uncertainties. We will show that these appear to be, ifanything, slight overestimates of the uncertainty.

The posterior pdf (equation 3) is calculated on an grid ofisochrones at metallicities as given in Table 2 and ages spacedby δlog10(τ /yr) = 0.04 for τ < 1 Gyr and δlog10(τ /yr) = 0.01 forτ > 1 Gyr. For each of these isochrones we take grid points in ini-tial mass M such that there is no band in which any magnitudechanges by more than 0.005 mag. We then evaluate the posterioron an informed grid in log AV and distance, which is centred onthe expected log AV from the prior at an estimated distance (giventhe observed and model J-band magnitude) and then the estimateddistance (given each log AV value evaluated).

Where stars have been observed more than once by RAVE, weprovide distance estimates for the quoted values from each spec-trum. We provide a flag ‘flag dup’ which is 0 if the spectrum is thebest (or only) one for a given star, as measured by the signal-to-noiseratio, and 1 otherwise. Where one wishes to avoid double countingstars one should only use rows where this flag is 0.2

2.1 Standard prior

For our standard results, we use the prior that was used for DR4and DR5. We do this for consistency, and because we find thatthis provides good results. The prior reflects some elements of ourexisting understanding of the Galaxy, at the cost of possibly biasingus against some results that run counter to our expectations (forexample, metal-rich or young stars far from the plane). In Section5 we consider alternative priors. Although the prior is described inBinney et al. (2014), we describe it here for completeness, and toenable comparisons with alternative priors considered.

The prior considers all properties in our model, and can be writtenas

P (model) = P (M, τ, [M/H], s, AV |l, b)= P (M) × P (AV | s, l, b) × P (s, [M/H], τ | l, b) (7)

2We have based this on the RAVEID number for each source. It is worthnoting that the cross-matching of stars is not perfect, and so despite our bestattempts to clean duplicate entries, there may be a few per cent of stars thatare in fact listed twice.



with the prior on initial mass being a Kroupa (2001) initial massfunction (IMF), as modified by Aumer & Binney (2009)

P (M) ∝

⎧⎪⎪⎨⎪⎪⎩

0 if M < 0.1 M�M−1.3 if 0.1 M� ≤ M < 0.5 M�,0.536M−2.2 if 0.5 M� ≤ M < 1 M�,0.536M−2.519 otherwise.

(8)

We describe extinction in terms of the value AV for the JohnsonV band, and, since extinction is necessarily non-negative, we takeour prior to be Gaussian in ln AV around an expected value whichvaries with the model star’s position in the Galaxy, ln AprV (s, l, b).

To find the expected value AprV (s, l, b) we start from an expectedvalue at infinity, AprV (∞, l, b), which we take from the Schlegel,Finkbeiner & Davis (1998) values of E(B − V), with a correctionfor high extinction sightlines following Arce & Goodman (1999)and Sharma et al. (2011), leaving us with

AprV (∞, l, b) = 3.1 × E(B − V )SFD {0.6 + 0.2

×[

1 − tanh(

E(B − V )SFD − 0.150.3

)]}. (9)

We then determine the expected extinction at a given distance sin the direction l, b, which is some fraction of the total extinctionalong that line of sight. We take this to be the fraction of the totalextinguishing material along that line of sight that lies closer thans in a 3D dust model of the Milky Way taken from Sharma et al.(2011). For details of the model see Binney et al. (2014).

As in Binney et al. (2014) we take the uncertainty in ln AV to be√2. We can then write the prior on AV to be

P (AV |s, l, b) = G(ln AV , ln(AprV (s, l, b)),√

2). (10)

Extinction varies between different photometric bands. For agiven extinction value AV, from Rieke & Lebofsky (1985) we takethe extinctions to be

AJ = 0.282AVAH = 0.175AVAKs = 0.112AV , (11)and, following from this, and using the results of Yuan, Liu & Xiang(2013), we have extinction in the WISE photometric bands of

AW1 = 0.0695AVAW2 = 0.0549AV . (12)

The other term in the prior is related to the probability of therebeing a star of a given τ , [M/H], and position. It also contains afactor of s2, to reflect the conical shape of the surveyed volume.3

The prior on distance, [M/H], and age can then be written as

P (s, [M/H], τ | l, b) ∝ s23∑

i=1NiPi([M/H]) Pi(τ ) Pi(r), (13)

where i= 1, 2, 3 correspond to a thin disc, thick disc, and stellarhalo, respectively, and where r is the Galactocentric position of thestar. We then have

3This factor was stated by Burnett & Binney (2010), but not directly notedby either Burnett et al. (2011) or Binney et al. (2014), who simply stated thedensity profile associated with the prior on position. This oversight meantthat Santiago et al. (2016) noted the absence of this factor as a differencebetween the Binney et al. (2014) values and their own, closely related,results. The factor of s2 was, however, used in all of these studies.

Thin disc (i = 1):

P1([M/H]) = G([M/H], 0, 0.2),P1(τ ) ∝ exp(0.119 τ/Gyr) for τ ≤ 10 Gyr,P1(r) ∝ exp

(− R

Rthind− |z|

zthind

); (14)

Thick disc (i = 2):

P2([M/H]) = G([M/H], −0.6, 0.5),P2(τ ) ∝ uniform in range 8 ≤ τ ≤ 12 Gyr,P2(r) ∝ exp

(− R

Rthickd− |z|

zthickd

); (15)

Halo (i = 3):

P3([M/H]) = G([M/H], −1.6, 0.5),P3(τ ) ∝ uniform in range 10 ≤ τ ≤ 13.7 Gyr,P3(r) ∝ r−3.39; (16)

where R signifies Galactocentric cylindrical radius, z cylin-drical height, and r spherical radius. We take Rthind = 2600 pc,zthind = 300 pc, Rthickd = 3600 pc, zthind = 900 pc. These values aretaken from the analysis of SDSS data in Jurić et al. (2008). Themetallicity and age distributions for the thin disc come from Hay-wood (2001) and Aumer & Binney (2009), while the radial densityof the halo comes from the ‘inner halo’ detected in Carollo et al.(2010). The metallicity and age distributions of the thick disc andhalo are influenced by Reddy (2010) and Carollo et al. (2010). Thehalo component tends towards infinite density as r → 0, so we applyan arbitrary cut-off for r < 1 kpc – a region which the RAVE sampledoes not, in any case, probe.

The normalizations Ni were then adjusted so that at the Solarposition, taken as R0 = 8.33 kpc (Gillessen et al. 2009), z0 = 15 pc(Binney, Gerhard & Spergel 1997), we have number density ratiosn2/n1 = 0.15 (Jurić et al. 2008), n3/n1 = 0.005 (Carollo et al. 2010).

3 C O M PA R I S O N O F D R 5 A N D T G A SPA R A L L A X E S

For RAVE DR5 the distance estimation used the 2MASS J, H,and Ks values, and the Teff, log g, and [M/H] values calculatedfrom RAVE spectra. The parallaxes computed were compared withthe parallaxes obtained by the Hipparcos mission (Perryman et al.1997), specifically those found by the new reduction of van Leeuwen(2007) for the ∼5000 stars common to both catalogues. The paral-laxes were compared by looking at the statistic

� =〈�sp

〉 − �ref√σ 2�,sp + σ 2�,ref

, (17)

where � sp and σ� , sp are the spectrophotometric parallax estimatesand their uncertainties, respectively. In Kunder et al. (2017) thereference parallax � ref and its uncertainty σ� , ref were from Hip-parcos, but henceforth in this paper they will be from TGAS. Anegative value of �, therefore, corresponds to an overestimate ofdistance from RAVE (compared to the reference parallaxes), anda positive value corresponds to an underestimate of distance. We



would hope that the mean value of � is zero and the standard de-viation is unity (consistent with the uncertainties being correctlyestimated).

Here, as in Kunder et al. (2017) we divide the stars into dwarfs(log g ≥ 3.5) and giants (log g < 3.5), and further subdivide dwarfsinto hot (Teff > 5500 K) and cool (Teff ≤ 5500 K). It is worth notingthat this means that main-sequence turn-off stars are likely to be putin the ‘dwarf’ category. In Fig. 2 we show a comparison betweenthe DR5 parallaxes and the TGAS parallaxes described by thisstatistic (which we call �DR5-TGAS in this case). The figures showkernel density estimates (KDEs; Scott 1992), which provide anestimate of the pdf of �DR5 for each group, along with finely binnedhistograms (which are used to give a sense of the variation aroundthe smooth KDE). These are generally encouraging for both cooldwarfs and giants, with a mean value that is close to zero (meaningthat any parallax, and therefore distance, bias is a small fraction ofthe uncertainty), and a dispersion that is slightly smaller than unity(implying that the uncertainties of one or both measurements areoverestimated).

For hot dwarfs there is a clear difference between the DR5 par-allaxes and the TGAS parallaxes. The mean value of � is 0.301,meaning that the systematic error in parallax is a significant frac-tion of the uncertainty, with the DR5 parallaxes being systematicallylarger than the TGAS parallaxes (corresponding to systematicallysmaller distance estimates from DR5).

The typical combined quoted uncertainty on the parallaxes for hotdwarfs is ∼1 mas, so this systematic difference is ∼0.3 mas, whichis comparable to the size of the colour-dependent and spatiallycorrelated uncertainties identified by Lindegren et al. (2016). It wastherefore not immediately obvious whether the difference seen hereis due to a systematic error with the DR5 parallaxes, or with theTGAS parallaxes.

However, we have indications from Kunder et al. (2017) thatthe effective temperatures found by the RAVE pipeline tend to beunderestimates for Teff � 5300 K. The effective temperatures deter-mined using the IRFM are systematically higher than those foundfrom the RAVE pipeline (Fig. 26; Kunder et al. 2017). If the effec-tive temperature used in the distance estimation is systematicallylower than the true value, then this will cause us to systematicallyunderestimate the luminosity of the star, and thus underestimate itsdistance (overestimate its parallax). Therefore, a systematic under-estimate of Teff by the RAVE pipeline can explain the differencewith the IRFM Teff values and the systematic difference with theTGAS parallaxes. This motivates us to investigate the IRFM tem-peratures in Section 4 for an improved estimate of Teff, and thusmore accurate distance estimates.

We can investigate this more closely by looking at how an averagevalue of �DR5 (which we write as 〈�DR5〉) varies with Teff for dwarfsor with log g for giants. In Fig. 3 we show the running average ofthis quantity in windows of width 200 K in Teff for dwarfs and 0.3dex in log g for giants. For reference we also include the numberdensity as a function of these parameters in each case.

The left-hand panel of Fig. 3 shows the value of 〈�DR5-TGAS〉(Teff)for dwarfs. As we expect, we see that for Teff � 5500 K we have aparallax offset of ∼0.3 times the combined uncertainty, which hasa small dip around 7400 K.4 The vast majority of what we termed‘cool dwarfs’ are in the temperature range 4600 � Teff < 5500 K,where TGAS and RAVE clearly agree nicely.

4The sharp edges are due to the fact that a relatively large number of sourcesare assigned temperatures very near to 7410 K, due to the pixelization pro-duced by the fitting algorithm – see Kordopatis et al. (2011).

Below ∼4600 K the value of 〈�〉(Teff) goes to very large values,corresponding to a substantial underestimate of distance by RAVEDR5. This was not clearly seen in Fig. 2 because there are very fewdwarfs in this temperature range. It is not clear what causes this,though it could occur if (1) there is a tendency to underestimatethe Teff for these stars, which is not something which has beennoted before; (2) stars with quoted logg values between the dwarfand giant branches have been given too high a probability of beingdwarfs by the pipeline, and/or (3) the pipeline assigns too low aluminosity to stars near this part of the main sequence – possiblybecause many of them are still young and perhaps still settling onto the main sequence (see Žerjal et al. 2017).

The right-hand panel of Fig. 3 shows the value of 〈�DR5〉(log g)for giants. In the range 2.2 � log g � 3.0 (which is a region with ahigh number of stars) we can see that the DR5 parallaxes more orless agree with those from TGAS. However, at high logg RAVE par-allaxes are on average larger than those from TGAS (correspondingto an underestimate of the luminosity), whereas at low log g RAVEparallaxes are on average smaller than those from TGAS (i.e. theluminosity is overestimated). We will discuss this difference in Sec-tion 4.1.

It is worth emphasizing that the effects we see here for low Teff orlow log g are not ones that we would simply expect to be caused bythe statistical uncertainties in the RAVE parameters (e.g. the starswith the lowest quoted log g values being only the ones scatteredthere by measurement error). The Bayesian framework compensatesfor exactly this effect, so the problem we are seeing is real.

4 U S I N G OTH E R R AV E DATA PRO D U C T S FO RDISTANCE ESTIMATION

We now look at how the difference between parallaxes derived fromRAVE and those from TGAS compare if we use Teff values derivedfrom the IRFM, rather than those derived from the spectrum directly.We also include WISE photometry in the W1 and W2 bands in bothcases (as discussed in Section 2).

Fig. 4 again shows the difference between the parallaxes wederive and those found by TGAS, divided into the same three cat-egories. We can see that the disagreement for hot dwarfs is signif-icantly reduced from that found for DR5, with a systematic off-set that is half that seen when using the spectroscopic Teff values.However, we can also see that the agreement between the two val-ues is now slightly less good than before for cool dwarfs andfor giants.

We can explore this in more detail by, again, looking at how theaverage value of � varies as we look at different Teff for all dwarfs. InFig. 5 we show how a running average, 〈�〉(Teff), varies for dwarfswhen we use the IRFM or the spectroscopic Teff values.5 It is clearthat whatever we choose as a Teff value, our parallax estimatesdiffer dramatically from those from TGAS for dwarfs with Teff �4600 K, but there are very few dwarfs with these temperatures. For4600 K�Teff � 5500 K the values found using the spectroscopicallydetermined Teff values are better than those found using the IRFMvalues, while for Teff � 5500 K the IRFM values are better. Evenusing the IRFM temperatures, the parallaxes found at Teff ∼ 6400 Kare still somewhat larger than those found by TGAS.

5Note that the 〈�〉 values using the spectroscopic Teff values are now notthose given in DR5, but new ones, found when we include the WISE pho-tometry. These prove to be very similar to those found by DR5.



Figure 2. Comparison of parallax estimates from RAVE DR5 and those from TGAS. We divide the stars into giants (log g < 3.5), cool dwarfs (log g ≥ 3.5and Teff ≤ 5500 K), and hot dwarfs (logg ≥ 3.5 and Teff > 5500 K) and provide pdfs of � (i.e. difference between spectrophotometric parallax and TGASparallax, normalized by the combined uncertainty, see equation 17) in each case. The red lines show the kernel density estimate of this pdf in each case, withthe finely binned grey histogram shown to give an indication of the variation around this smooth estimate. The black dashed line is a Gaussian with a mean 0and standard deviation of unity. The means and standard deviations shown in the top right are for stars with −4 < �DR5 < 4, to avoid high weight being givento outliers. Positive values of � correspond to parallax overestimates (i.e. distance or luminosity underestimates).

Figure 3. Running average of � (i.e. difference between spectrophotometric parallax and TGAS parallax, normalized by the combined uncertainty; seeequation 17) as a function of Teff for dwarfs (left lower) and log g for giants (right lower), comparing DR5 values to those from TGAS. The running averagesare computed for widths of 200 K and 0.3 dex, respectively. The plot also shows the number density as a function of Teff and log g, respectively, for reference.Means are only calculated for stars with −4 < �DR5-TGAS < 4. Note that positive values of � correspond to parallax overestimates (i.e. distance or luminosityunderestimates).

4.1 Giants

We can now turn our attention to the giant stars. When we simplydivide the stars into dwarfs and giants – as was done with Hipparcosparallaxes by Binney et al. (2014) and Kunder et al. (2017), and withTGAS parallaxes in Figs 2 and 4 of this study – any biases appearsmall. However, when we study the trend with logg, as in Figs 3and 6, we see that while the stars with log g � 2.2 have RAVE par-allaxes that are very similar to those from TGAS (with a moderateoverestimate for log g < 3), the stars with lower log g values haveRAVE parallaxes which seem to be systematically underestimated(corresponding to distance overestimates).

We can understand how this may have come about if we lookat the comparison of the RAVE log g values with those found byGALAH (Martell et al. 2017) or APOGEE (Wilson et al. 2010) forthe same stars – as presented in figs 17 and 19 of Kunder et al.(2017). In both cases there appears to be a trend that the othersurveys find larger log g values for stars assigned RAVE log g �

2. A systematic underestimate of the log g values of these starswould lead to exactly this effect. In Section 4.1.1 we will look atthe asteroseismic re-calibration of RAVE log g found by Valentiniet al. (2017), which also suggests that these log g values may beunderestimated.

It is important to note that these low log g stars are intrinsicallyluminous, and therefore those observed by RAVE tend to be distant.This means they have relatively small parallaxes, and so the quotedTGAS uncertainties are a large fraction of true parallax, while thosefrom RAVE are relatively small. Fig. 7 illustrates this point byshowing the median parallax and uncertainty for each method as afunction of log g.

A consequence of this is that the combined parallax uncertaintyused to calculate � is dominated by that from TGAS. We illustratethis in Fig. 8, which shows the median value of the alternativestatistic (�IRFM − �TGAS)/σ�,IRFM, where � IRFM is the parallaxestimate using the IRFM Teff value, and σ�,IRFM is the corresponding



Figure 4. Comparison of parallax estimates from RAVE with temperatures taken from the IRFM and parallax measurements from TGAS. This plot showsthe same statistics as in Fig. 2, and again we divide the stars into giants (log g < 3.5), cool dwarfs (log g ≥ 3.5 and Teff, IRFM ≤ 5500 K), and hot dwarfs (logg≥ 3.5 and Teff, IRFM > 5500 K) and provide pdfs of � (equation 17) in each case – positive values of � correspond to parallax overestimates (i.e. distance orluminosity underestimates). The main difference we can see is that the parallax estimates for hot dwarfs are substantially improved.

Figure 5. As Fig. 3 (left-hand panel), this is a running average of � as afunction of Teff for dwarfs (log g ≥ 3.5), but here we are using Teff valuesdetermined by the IRFM (blue) or from the RAVE spectra (green). Again,the plot also shows the number density of dwarfs as a function of Tefffor reference. Use of the IRFM temperatures reduces the bias seen for hotdwarfs.

Figure 6. As Fig. 3 (right-hand panel), this is a running average of � as afunction of logg for giants (log g < 3.5), but here we are using Teff valuesdetermined by the IRFM (blue) or from the RAVE spectra (green). Again,the plot also shows the number density as a function of log g, respectively,for reference. Means are calculated for stars with −4 < � < 4.

Figure 7. Median parallax (solid line) and median parallax uncertainty(shaded region) for the RAVE pipeline using IRFM Teff values (blue) andTGAS (red) as a function of log g. The quoted parallax uncertainty fromRAVE becomes much smaller than that from TGAS as log g becomes small.This means that when we use the TGAS parallaxes to improve the distanceestimates, they will have little influence at the low log g end.

Figure 8. Running average of (�IRFM − �TGAS)/σ�,IRFM as a function oflog g for giants (log g < 3.5) – this statistic is similar to � used elsewhere,but does not include the TGAS uncertainty. It therefore shows the typicalsystematic offset of the RAVE parallax estimates as a function of the quoteduncertainty. For the lowest log g, the two values are comparable.



uncertainty.6 This shows that the systematic error for the lowest log gstars is comparable to the quoted statistical uncertainty.

This also means that when we include the TGAS parallaxes inthe distance pipeline for these objects, it will typically have a ratherlimited effect, and so the bias that we see here will persist.

4.1.1 Asteroseismic calibration

The log g values given in the main table of RAVE DR5 have aglobal calibration applied, which uses both the asteroseismic log gvalues of 72 giants from Valentini et al. (2017) and those of theGaia benchmark dwarfs and giants (Heiter et al. 2015). This leadsto an adjustment to the raw pipeline values (which were used inRAVE DR4, so we will refer to them as log gDR4) such that

log gDR5 = log gDR4 + 0.515 − 0.026 × log gDR4− 0.023 × log g2DR4. (18)

A separate analysis was carried out by Valentini et al. (2017) whichfocused only on the 72 giants with asteroseismic log g values,which are only used to recalibrate stars with dereddened colours0.50 < (J − Ks)0< 0.85 mag, and they found that for these stars amuch more drastic recalibration was preferred, with the recalibratedlog g value being

log gAS = log gDR4 − 0.78 log gDR4 + 2.04≈ 2.61 + 0.22 × (log gDR4 − 2.61). (19)

This has the effect of increasing the log g values for stars in the redclump and at lower log g – thus decreasing their expected luminos-ity and distance, and increasing their expected parallax. It has theopposite effect on stars at higher log g. It is clear, therefore, thatthis recalibration is in a direction required to eliminate the trendin � with log g for giants seen in the right-hand panel of Fig. 3. Itis also worth noting that Kunder et al. (2017) compared log gAS toliterature values and found a clear trend in the sense that log gASwas an overestimate for stars with literature log g < 2.3, and anunderestimate for literature log g > 2.8.

In Fig. 9 we show � as a function of log gDR5 for stars using therecalibrated log gAS values given by Valentini et al. (2017) (alongwith those when using the DR5 log g values for reference). Weuse the DR5 log g value on the x-axis to provide a like-for-likecomparison, and the grey region in Fig. 9 is equivalent to the range2.3 < log gAS < 2.8. It is clear that the asteroseismically calibratedlog g values improve the distance estimation for stars with lowlog g values – even beyond the range of log g values where theselog g values disagree with other external catalogues (as found byKunder et al. 2017) – though it should be noted that these stars [with0.50 < (J − Ks)0< 0.85 mag] represent a small fraction of the starswith these low logg values.

However, for gravities greater than log gDR5 � 2.5 (which isthe point where log gAS = log gDR5), the asteroseismic calibrationmakes the log g values significantly worse in the sense that thespectrophotometric parallaxes are underestimates (i.e. the distancesare typically overestimated). Inspection of the comparison of RAVEDR5 log g values to those from GALAH or APOGEE in Kunderet al. (2017) appears to indicate that those with log gDR5 ≈ 3 aresplit into two groups (one with higher log g found by the othersurveys, one with lower) – i.e. these are a mixture of misidentified

6Because the TGAS uncertainty is far smaller than the RAVE uncertaintyfor dwarfs, the equivalent plot for them is very similar to that in Fig. 5.

Figure 9. As Fig. 6 this is a running average of � as a function of log gDR5for giants where the log g values used come from the main DR5 calibration(blue; equation 18) or the asteroseismic calibration (red; equation 19). Notethat the x-axis gives the DR5 log g value in each case – this is to enable aside-by-side comparison. In both cases we have used Teff values determinedby the IRFM. The grey region indicates the range in log g over which theasteroseismic calibration appears to work reasonably well for the referencestars considered by Kunder et al. (2017). The running averages are computedfor over a width of 0.3 in log g. The plot also shows the number density as afunction of log g, respectively, for reference. Means are calculated for starswith −4 < � < 4. Using the asteroseismically calibrated logg values forstars clearly improves the distance estimates for log gDR5 � 2.5, which isthe point where the two values are equal, but makes them worse for log gDR5� 2.5.

dwarfs/subgiants and giants. The asteroseismic calibration is blindto this difference, and it seems likely that it does a reasonable job ofcorrecting the logg values for the giants, at the cost of dramaticallyunderestimating the log g values for the dwarfs/subgiants at thesame loggDR5.

The Valentini et al. (2017) catalogue comes with an entry‘flag 050’ which is true if the difference between log gDR5 andlog gAS is less than 0.5, and it is recommended that only stars withthis flag are used. This sets an upper limit of loggDR5 � 3.5 forsources where the asteroseismic calibration can be applied. Ourwork here implies that the asteroseismic calibration should not beused for sources with log gDR5 � 2.7.

4.2 Outliers

We have ∼1000 stars for which the quoted parallaxes from RAVEand TGAS differ by more than 4σ . We will refer to these as ‘out-liers’. We would only expect ∼12 such objects if the errors wereGaussian with the quoted uncertainties. In Fig. 10 we show pdfsindicating how these stars are distributed in quoted Teff, IRFM, log g,and [M/H]. They cover a wide range of these parameters, and noclear problematic area is evident. They do tend to have relativelylow Teff values, and constitute a relatively large fraction of starswith quoted [M/H] values towards either end of the full range.

We also show the distribution of these stars in terms of S/N, andwe can see that while they tend to have relatively low S/N values,they are certainly not limited to such stars. We have also looked at thevalues of the AlgoConv quality flag, which is provided with RAVEparameters, and find that the outliers are indicated as unreliablearound the same rate as the rest of the sources. Around 26 percent of the outliers have flags 2 or 3, which indicate that the stellarparameters should be used with caution, as compared to ∼23 percent of all other sources, which suggests that this is not the problem.



Figure 10. Distributions of the quoted parameters of the ∼1000 stars thatare outliers in the sense that they have |�| > 4 (blue lines), and of all starsin the study, for reference (green lines). The plots are pdfs (so the area isnormalized to 1 in all cases) produced using a kernel density estimate. Thedistributions shown are in Teff, IRFM (top left), log gDR5 (top right), [M/H](bottom left), and S/N (bottom right). The outliers cover a wide range of theseparameter spaces, and do not come from any clearly distinct population.

There is also no indication that they are particularly clustered onthe sky.

There is some indication that the outliers tend to be problematicsources as labelled by the flags from Matijevič et al. (2012), whichare provided with DR5. These flags are based on a morphologicalclassification of the spectra, and can indicate that stars are peculiar(e.g. have chromospheric emission or are carbon stars) or that thespectra have systematic errors (e.g. poor continuum normalization).∼20 per cent of the outliers are flagged as binary stars, and ∼35per cent are flagged as having chromospheric emission (comparedto ∼2 per cent and ∼6 per cent of all sources, respectively). Sim-ilarly, ∼40 per cent of the outliers are in the catalogue of starswith chromospheric emission from Žerjal et al. (2013, 2017). Thechromospheric emission can only have affected the RAVE distanceestimates. However, binarity can affect either the RAVE distance(by affecting the parameter estimates and/or observed magnitudes)or the TGAS parallaxes (by altering the star’s path across the sky,thus changing the apparent parallax).

4.3 Metallicity

Finally, we can look at the variation of � with more than one stellarparameter. In Fig. 11 we show the variation of � in the Hertzsprung–Russell (HR) diagram (Teff against log g) for all stars. We also showthe variation of � in the [M/H]–Teff plane for dwarfs and the [M/H]–logg plane for giants. In all cases we just show the statisticswhen we use the IRFM temperatures.

The HR diagram shows some areas where RAVE parallaxes ap-pear to be particularly discrepant. We had already seen that low-temperature dwarfs (Teff, IRFM � 4500 K) have overestimated paral-laxes. The sources with Teff, IRFM∼ 5000 Kand loggDR5∼ 4.2 haveunderestimated parallaxes. These sources are between the dwarf and

subgiant branches, and it appears that they are typically assignedtoo high a probability of belonging to the subgiant branch. Thesewill be greatly improved when we include the TGAS parallax inour estimates. Sources at the upper edge of the giant branch (highquoted Teff for their quoted log g) also have very small RAVE paral-laxes compared to those from TGAS, but these are a small fractionof giant stars.

There are no clear trends with metallicity for giants. For thedwarfs it is perhaps notable that there are significant parallax un-derestimates for metal-poor stars at Teff∼ 5200 K and parallax over-estimates for both unusually metal-poor and metal-rich stars atTeff ∼ 6200 K. Again these do not comprise a particularly largefraction of all sources, and will be corrected when we include theTGAS parallax in our estimates. It is worth noting that selection ef-fects mean that the more metal-poor stars (which tend to be furtherfrom the Sun in the RAVE sample) are likely to be higher temper-ature dwarfs, and (particularly) lower log g giants, and this affectsany attempts to look at variation of � with metallicity independentof the other stellar parameters.

Since the most metal-poor stars tend to be cool giants which,as we have noted, are assigned distances in our output that aresystematically too large, a sample of our stars which focusses onthe metal-poor ones will suffer from particularly serious distanceoverestimates. Any prior which (like our standard one) assumesthat metal-poor stars are the oldest will have a similar overestimatefor the stars that are assigned the oldest ages in the sample. Note,however, that the age estimates we provide are found using a priorwhich assumes no such age–metallicity relation (see Section 5.1),so the most metal-poor stars are not necessarily assigned the oldestages in our catalogue.

4.4 Which to use?

It is clear that adopting the IRFM temperature estimates improvesthe distance estimates for stars that have Teff, Spec > 5500 K. Useof the IRFM temperatures does make the problems at low log gsomewhat worse than they already were, but this is a smaller effect.We feel that switching from one temperature estimate to another atdifferent points in the HR diagram would be a mistake, so we usethe IRFM temperature in all cases. For ∼5000 sources there is noIRFM Teff available, so we do not provide distance estimates.

For sources recognized as outliers (|�| > 4) we assume thatthe RAVE parameters are unreliable, in the published cataloguethese are flagged, and we provide distances estimated using onlyTGAS parallaxes and the 2MASS and WISE photometry. Similarly,we recognize that there is a systematic problem with dwarfs atTeff < 4600 K, so for these stars we exclude the RAVE Teff andlog g from the distance estimation, and add an (arbitrary) 0.5 dexuncertainty in quadrature with the quoted RAVE uncertainty onmetallicity.

We have seen that sources with log gDR5 < 2.0 show a system-atic difference between our parallax estimates and those found byTGAS. This is probably due to a systematic underestimate of log gfor these stars by RAVE. We will determine distances to these starsin the same way as to the others, but they will be flagged as prob-ably unreliable. While the asteroseismic recalibration clearly helpsfor these stars, it is not helpful at high log g, and is applicable to adwindling fraction of sources as we go to lower log g. We thereforedo not attempt to use this recalibration in our distance estimates,though it certainly indicates the direction we must go to improvethe RAVE log g estimates.



Figure 11. Median values of �, using IRFM temperatures, as a function of the stellar parameters Teff, log g, and [M/H]. Pixel sizes are adapted such that thereis never fewer than 10 stars in a pixel for which we show the median. For the variation with metallicity we have, as before, divided the stars into dwarfs andgiants, to show the more relevant parameter in each case. The grey areas contain very few stars. Density contours are shown as a guide to the location of themajority of the sources in these plots (this shows signs of the pixelization of these parameters produced by the fitting algorithm used in the RAVE spectroscopicpipeline).

5 A LT E R NAT I V E PR I O R S

It would be very troubling if our results were strongly dependenton our choice of prior. We therefore explore the effect of our priorby considering alternative forms. We will call our standard prior‘Standard’, and describe the differences from this prior. We considerfour main alternative forms:

(i) ‘Density’ prior. As Standard, except that we set the prior on[M/H] and τ to be uniform, with a maximum age of 13.8 Gyr.The minimum and maximum metallicities are effectively set by theisochrone set used (Table 2).7 This leaves the density profile, IMF,and dust model unchanged.

(ii) ‘Age’ prior. As Standard, except that the age prior is the samefor all components and simply reflects the assumption that the starformation rate has declined over time, following the same functionalform as for the thin disc in the Standard prior, i.e.

P (τ ) ∝ exp(0.119 τ/Gyr) for τ ≤ 13.8 Gyr. (20)(iii) ‘SB14’ prior. As Standard, except that we set the prior on

[M/H] and τ identically for all components, following Schönrich &Bergemann (2014). This is uniform in [M/H] over the metallicityrange set by the isochrones, such that

P (τ | [M/H]) ∝

⎧⎪⎨⎪⎩

0 if τ > 13.8 Gyr1 if 11 Gyr ≤ τ ≤ 13.8 Gyrexp

[(τ−11 Gyr)στ ([M/H])

]if τ ≤ 11 Gyr,

(21)

where

στ =⎧⎨⎩

1.5 Gyr if [M/H] < −0.9(1.5 + 7.5 × 0.9+[M/H]0.4

)Gyr if − 0.9 ≤ [M/H] ≤ −0.5

9 Gyr otherwise.

(22)

(iv) ‘Chabrier’ prior. As Standard, except that we use a Chabrier(2003) IMF rather than a Kroupa (2001) IMF, where following

7It is possible to remove this limitation, under the assumption that the stellarmodels do not change much at lower or higher metallicities, but the effectis limited, and it is not implemented here.

Romano et al. (2005) we take

P (M) ∝

⎧⎪⎨⎪⎩

0 if M < 0.1 M�AcM exp

(log10 M−log10 Mc

σc

)2if 0.1 M� ≤ M < M�,

Bc M−2.3 otherwise.(23)

In Fig. 12 we compare the values of � that we derive underall of these priors, in each case using the sets of input parametersdescribed in Section 4.4, and excluding sources where we ignorethe RAVE parameters.

It is clear from the left-hand panel of Fig. 12 that the priors makea very limited difference for the dwarfs, except at the low Teff end,where contamination by giants is becoming more important.

The right-hand panel of Fig. 12 shows that for giants, a priorthat is uniform in both [M/H] and stellar age – i.e. the Densityprior – provides even worse results for the low log g giants than theStandard prior. The other priors provide very similar results to oneanother at low log g, but differ somewhat at the higher log g end –the two priors, where P([M/H]) is a function of position (Standardand Chabrier), tend to have lower � values, i.e. greater distances tothese stars derived from RAVE.

We have also explored the effect of changing the power-law slopeof the halo within our Standard prior (equation 16) to either P3(r) ∝r−3.9 or P3(r) ∝ r−2.5 (compared to the usual r−3.39). The resultswere essentially indistinguishable from those using the Standardprior, even if we isolate the metal-poor stars. Similarly, a decreaseof 50 per cent for the thin and thick disc scale heights has almost noeffect – the mean and standard deviations of the � values for a givenpopulation of stars (as shown in, e.g. Fig. 4) change by ∼0.001 atmost.

5.1 Choice of prior

In the interests of consistency with past studies, we use the Standardprior when producing our distance estimates. However, it is clearthat this choice of prior imposes a strong relationship between ageand metallicity. Therefore, we also provide age estimates (Section8) using our ‘Age’ prior. The results presented in this section makeit clear that results using this prior are roughly as reliable as thosefrom our Standard prior, at least in terms of typical parallax error.



Figure 12. As Fig. 3, this is the running average of � as a function of Teff for dwarfs (left lower) and log g for giants (right lower) when using the alternativepriors described in Section 5. In general, the RAVE distance estimates are reasonably robust to a change of prior.

6 U S I N G R AV E PA R A L L A X E S TO L E A R NA B O U T TG A S

In Section 4 we used the TGAS parallaxes to investigate the RAVEdistance estimation, but we can turn this around and use the RAVEdistance estimation to learn about TGAS. TGAS is an early releaseof Gaia data and is therefore expected to contain strong systematicerrors (Gaia Collaboration 2016b; Lindegren et al. 2016). Variousstudies have looked at these systematic errors (including the Gaiaconsortium itself: Arenou et al. 2017), by comparison to distancesderived for RR Lyrae stars (Gould et al. 2016), red clump stars(Davies et al. 2017; Gontcharov & Mosenkov 2017), or eclipsingbinaries (Stassun & Torres 2016) or, in the case of Schönrich &Aumer (2017), using a statistical approach based on the correla-tions between velocity components produced by distance errors.Our approach allows us to study a large area in the southern skyusing many sources, spanning a wide range in colour, without anyassumptions about kinematics.

In Fig. 13 we plot the average difference between the TGAS par-allax and that from this study, binned on the sky. Zonal differencesare unlikely to be produced by any particular issues with the RAVEdistance estimation, but may be related to the way in which thesky has been scanned by Gaia. We can clearly see a stripe showinga substantial difference at l∼ 280◦, which corresponds to a stripenear the ecliptic pole, as can be seen when this diagram is shownin ecliptic coordinates. A similar figure was shown in Arenou et al.(2017, fig. 28), using the RAVE DR4 parallax estimates, where thisfeature was attributed to the ‘ecliptic scanning law followed earlyin the mission’, and it was noted that a corresponding feature canbe found in the median parallaxes of quasar sources. This is alsolikely to be related to the anomaly reported by Schönrich & Aumer(2017).

We can also look again at the width of the distribution of �.As we have seen already, the width of the distribution of �, whencomparing TGAS and DR5, is less than unity. In Fig. 14 we showthis width for all stars in our new RAVE-only parallax estimates, andit is again less than unity. This indicates that the uncertainties of oneor other measurements have been overestimated. When we dividethe distribution by quoted TGAS parallax uncertainty (Fig. 15) wecan see that the problem is particularly acute for sources with smallquoted TGAS uncertainties.

As discussed in Lindegren et al. (2016), uncertainties in the finalTGAS catalogue are designed to be conservative, and have been‘inflated’ from the formal uncertainties derived internally. This was

Figure 13. Absolute difference between TGAS parallaxes and the newRAVE-only parallax estimates averaged (median) in bins on the sky in anAitoff projection, shown in Galactic coordinates (l, b, upper) and eclipticcoordinates (λ, β, lower – note that we have placed λ = 180◦ at the centreof this plot to clearly show the feature). In each plot the grey area is wherethere are few or no stars. The clearest feature is the patch near l ∼ 280◦,b ∼ 0◦ where TGAS parallaxes appear to be systematically larger than thosefrom RAVE. When looked at in ecliptic coordinates this area can be seen torun from ecliptic pole to ecliptic pole, and is therefore likely to be related toGaia’s scanning law (Arenou et al. 2017).

to take account of uncertainties that are not allowed for in the for-mal calculation (such as contributions from uncertainties in Gaia’scalibration and attitude). The scheme used was derived from a com-parison to the (independent) Hipparcos parallaxes, and the quoteduncertainties were determined from the formal uncertainties usingthe formula

σ 2�,TGAS = a2ς2�,TGAS + b2, (24)where ς� , TGAS is the formal parallax error derived internally,a = 1.4 and b = 0.2 mas.



Figure 14. Distribution of � for all stars using the new RAVE-only parallaxestimates compared to TGAS. The standard deviation is less than unity,implying that the uncertainties of at least one of the parallax estimates havebeen overestimated.

Gould et al. (2016) looked at the reported parallaxes of RR Lyraestars in TGAS, and used the known period–luminosity relationshipfor these stars to provide an independent estimate of the uncer-tainties in parallax. They found that for these sources a = 1.1,b = 0.12 mas provides a better description of the true TGAS un-certainties, and therefore recommended that the TGAS parallaxestimates should be reduced to a value σ� , TGAS, sc given by theformula

σ 2�,TGAS,sc = α2σ 2�,TGAS − β2 (25)with α = 0.79 and β = 0.10. They investigated this by looking athow the sum of values of (their equivalent to) �2 varied as theyincreased the number of values that they summed over (ordered bynominal parallax uncertainty). This was done in the expectation thatit should increase linearly with slope unity. For ease of plotting weconsider the closely related statistic

χ2red,n =1

n

n∑i

�2i , (26)

where the sum is over the n sources with the lowest quoted TGASuncertainty. This should have a constant value of unity as we sumover increasing numbers of sources.

In Fig. 16 we show that, if we use the quoted uncertainties for bothRAVE and TGAS, χ2red remains smaller than unity for all stars. If weuse the prescription from Gould et al. (2016) then we come closerto unity when we consider all stars, but χ2red is clearly less than unitywhere the sum is over the stars with lower σ� , TGAS. This suggeststhat the Gould et al., prescription gives uncertainties that are stilloverestimated for stars with small σ� , TGAS and underestimated forthose with large σ� , TGAS.

Fig. 16 also shows two alternative scenarios. We show χ2red cor-responding to the best values of α and β (assuming that the RAVEuncertainties are correct), which are α = 0.95 and β = 0.20, whichcorresponds to b= 0. Even when we do this (i.e. set the minimumuncertainty from TGAS to zero), the combined uncertainty for thestars with the lowest TGAS uncertainties is clearly too large. Wetherefore also consider the effect of arbitrarily reducing the RAVEuncertainties according to the formula

σ�,RAVE,sc = γ σ�,RAVE, (27)

and find that a value of γ = 0.86 (while keeping the quoted TGASuncertainties) produces results that are roughly as good as the re-sults we find when deflating the TGAS uncertainties. It is worthnoting that, like the TGAS uncertainties, the RAVE stellar param-eter uncertainties were designed to be conservative (Kunder et al.2017).

It is possible that the RAVE uncertainties tend to be overstated,particularly if the quoted external uncertainty estimates are over-stated for most stars. While it certainly would not produce a sys-tematic overestimate that was well described by equation (27), itcould affect our estimates in a more complicated and subtle way.We are therefore not in a position to determine for sure whether itis the RAVE uncertainties or TGAS uncertainties (or both) that areoverestimated. We would note that a comparison of DR5’s parallaxestimates to those from Hipparcos did not suggest underestimateduncertainties in either instance (Kunder et al. 2017, fig. 25). Weadd that the dispersion in � is smaller than unity for both giantsand dwarfs, considered independently. We conclude that our resultsare consistent with the TGAS uncertainties being underestimated,though probably not in quite the same way as the prescription ofGould et al. (2016). We will not attempt to correct for any overesti-mates of uncertainty when calculating the combined RAVE+TGASestimates below.

7 C OMBINED DISTANCE ESTIMATES

A fundamental element of Bayesian analysis is the updating of theprobability of a hypothesis (for example, the hypothesized distanceto a star) as more evidence becomes available. TGAS parallaxesprovide new evidence regarding these distances, so we are requiredto take it into account when determining the distances. We can thinkof this as either an additional piece of input data, or as a prior onparallax for each star (in addition to the prior on distance impliedby equation 13) – the two statements are equivalent.

In previous sections we have investigated the properties of theRAVE distance pipeline in the absence of TGAS parallaxes, anddeveloped an understanding of the problems with each data set. Wenow incorporate the new evidence from these parallax measure-ments to obtain more accurate distance estimates than either canprovide in isolation. We do this by including them in the set ofinputs (Oi, σ i) in equation (3). It can be expected that the impact ofthe TGAS parallaxes will be greatest at Teff values below the turn-off where there is serious ambiguity whether a star is on the mainsequence or ascending the giant branch – an uncertainty which isreflected in the bimodal pdfs which we are forced to represent usingmulti-Gaussian fits (equation 4).

We have seen that the parallax estimates (from RAVE alone) forstars with log g values less than ∼2.0 appear to be particularly biasedin the sense that they are systematically lower than those found byTGAS. It is very likely that this is due to the RAVE log g values inthis range being systematically underestimated, as is also suggestedby a comparison to the log g values found by GALAH or APOGEEsurveys for the same stars. We noted in Section 4.1 that the TGASparallax uncertainties for these stars are significantly larger thanthose found from the RAVE distance pipeline. Therefore, we cannotexpect that our distance estimates for these stars are significantlyde-biased by including the TGAS parallax in the estimate.

In Fig. 17 we show the average value of � (again as a functionof Teff, IRFM for dwarfs and log gDR5 for giants) for the combinedparallax estimates, with the RAVE-only distance estimates (alsousing the IRFM temperatures) shown for comparison. One must bevery careful not to overinterpret these plots for several reasons (e.g.



Figure 15. Distribution of � using the new RAVE-only parallax estimates, separated into bins by TGAS parallaxes � T. The standard deviation is lessthan unity in each case, but increases as � T increases. This could be because the RAVE uncertainties are consistently overestimated, or because the TGASuncertainties are particularly overestimated for the smallest uncertainties.

Figure 16. The cumulative reduced χ2 of the TGAS parallax measure-ments with the new RAVE-only parallax estimates (equation 26) when theuncertainties of one set or the other have revised downwards. The differ-ent lines correspond to the original values (dark blue), the correction fromGould, Kollmeier & Sesar (2016, green; α = 0.79 and β = 0.10 in equation25), the best correction available when only ‘deflating’ RAVE uncertainties(red; γ = 0.86 in equation 27), and the best available only adjusting TGASuncertainties (light blue; α = 0.95 and β = 0.20)

the RAVE+TGAS parallaxes are obviously not independent of theTGAS ones; the uncertainties for RAVE+TGAS, which enter thecalculation of �, are generally much smaller than those of RAVEalone), but they clearly indicate that the difference at low logg valuesis not removed when we include the TGAS parallax information.

7.1 Improvement

Including the TGAS parallaxes in our distance estimation inevitablyleads to an improvement in the formal uncertainties. From the dis-cussion of the previous sections, we can claim with some confidencethat, outside a few regions of parameter space (e.g. low log g, thestripe near the ecliptic pole), the combination does not introducesignificant systematic errors into one data set or the other.

We can make a naı̈ve estimate of how the uncertainties willdecrease when we combine the two data sets by approximating thatthe uncertainties from the RAVE-only distance pipeline (σ� , Sp) areGaussian, in which case we have a new expected uncertainty inparallax σ� , exp given by

1/σ 2�,exp = 1/σ 2�,Sp + 1/σ 2�,TGAS. (28)Because the RAVE uncertainties are significantly non-Gaussian, wedo significantly better than this in some regions of the HR diagram.

This can be seen in Fig. 18, which shows the parallax uncertaintywe find divided by that which we would naively expect. This isalso reflected in the reduced number of stars for which the multi-Gaussian representations are required to describe the distance pdf(lower panel of Fig. 18).

In Fig. 19 we show how the fractional distance uncertaintyvaries over the HR diagram, both with and without TGAS par-allaxes. It is clear that the main improvement is for dwarfs, and forstars in the regions of the HR diagram where parallax informationcan break uncertainties regarding whether a star is a giant or adwarf.

When we include TGAS parallaxes, the median fractional dis-tance uncertainty (excluding stars with log gDR5< 2.0) falls to 15 percent, from 31 per cent using spectrophotometric information alone.For dwarfs the median uncertainty is just 10 per cent, while for gi-ants it is 19 per cent. The full pdfs of fractional distance uncertaintyare shown in Fig. 20.

The improvement over TGAS alone is shown in terms of paral-lax uncertainty in Fig. 21. In this case it is the giants for whichthe greatest improvement is found (again excluding stars withlog gDR5< 2.0). The median TGAS uncertainty is 0.32 mas for eithergiants or dwarfs, while the median uncertainty for RAVE+TGAS is0.20 mas for giants and 0.24 mas for dwarfs.

Using our combined estimates and the TGAS proper motions,we can convert this distance uncertainty into a velocity uncertainty.We take a simple Monte Carlo approach to do this – for each starwe sample from the multi-Gaussian pdf in distance modulus, andfrom Gaussians in proper motion and radial velocity with the quoteduncertainties. We again assume that the Sun is 8.33 kpc from theGalactic Centre and 15 pc from the Galactic plane. If we characterizethe resulting pdf in terms of a median value and a standard deviation(i.e. uncertainty) in each Galactocentric velocity component, we getthe distribution of uncertainties shown in Fig. 22. The introductionof TGAS parallaxes to our distance estimates improves the velocityaccuracy by, on average, ∼40 per cent in each direction.

Finally, we would like to estimate how we could correct ourdistance estimates to be unbiased. Since we do not know the truevalues we will do this under the assumption that the TGAS values areunbiased. We make the further approximation that – at a given Teffvalue for dwarfs or log g value for giants – we can simply multiplyall our RAVE+TGAS parallaxes by a correction factor corr� suchthat they are unbiased. For values of corr� ≈ 1 it follows thatthe equivalent factor for distances is corrs ≈ 2 − corr� . We findthe value of corr� by requiring that our statistic 〈�〉 is zero if wecompare corr� � RAVE + TGAS and � TGAS.

Fig. 23 shows the value of corr� we find as a function of Teffvalue for dwarfs and log g for giants. The dwarfs require systematic



Figure 17. The variation of the average value of � (equation 17) as a function of Teff for dwarfs (log g ≥ 3.5, left) and as a function of log g for giants(log g < 3.5, right). The Teff values come from the IRFM. In blue and labelled RAVE+TGAS we show our combined parallax estimates – we also show theRAVE-only estimates (using IRFM Teff values, in green and labelled RAVE, and as shown in Figs 5 and 6) to guide the eye. For low log g giants, TGASparallax uncertainties are too large to have a significant effect on the bias seen in RAVE.

Figure 18. The top panel shows the variation over the HR diagram of theratio of the actually quoted uncertainty on the parallax when combiningTGAS and RAVE data and the expected parallax uncertainty (equation 28)assuming Gaussian uncertainties. The Teff values come from the IRFM. Inthe region between the dwarf and giant branches and in the red clump theimprovement on naive expectations is particularly clear. The lower panelsprovide an explanation: they show the number of Gaussian componentsrequired to represent the pdf in distance modulus (equation 4) without TGASparallaxes (left) and with them (right). Without the TGAS parallaxes werequire a multi-Gaussian representation in ∼45 per cent of cases, whereaswith TGAS we only need it in ∼23 per cent of cases.

changes of less than 1 per cent in parallax (or distance) for all butthe hottest stars. The giants seem to require systematic changes ofmore than 10 per cent in parallax at logg< 2.0, up to around 35per cent at the lowest logg values. For these low log g stars, theapproximation corrs ≈ 2 − corr� becomes poor.

Figure 19. Average fractional distance uncertainties across the HR diagramwhen we ignore TGAS (left) and when we use the TGAS parallax informa-tion (right). The improvement is particularly dramatic for cooler dwarfs andstars with Teff ∼ 6000 K, log g ∼ 2.5. The Teff values come from the IRFM.For low log g giants, the inclusion of TGAS parallaxes has little effect.

Figure 20. Fractional distance uncertainties for sources when we ignoreTGAS parallaxes (upper panel) and when we use TGAS parallaxes (lowerpanel). In each case we show the pdfs for all sources (black), and separateones for giants (2.0 < log g < 3.5, red) and dwarfs (log g ≥ 3.5, blue). Thedashed lines show the median values in each case (0.33 and 0.16 withoutTGAS and with TGAS, respectively) for all stars (i.e. 51 per cent smallerwith TGAS), 0.36 and 0.20 for giants (44 per cent smaller), and 0.31 and0.10 for dwarfs (66 per cent smaller).



Figure 21. Parallax uncertainties when using the RAVE pipeline withTGAS parallaxes. The dotted curve is the pdf for all stars using just TGAS.The solid lines show the pdfs for all sources (black), and separate ones forgiants (2.0 < log g < 3.5, red) and dwarfs (log g ≥ 3.5, blue). The dashedlines show the median values in each case which can be compared to themedian TGAS uncertainty for these stars, which is 0.32 mas (essentially in-dependently of whether stars are dwarfs or giants). This median is 0.25 masfor all stars (24 per cent smaller than TGAS), 0.15 mas for giants (54 percent smaller), and 0.29 mas for dwarfs (9 per cent smaller).

Figure 22. Velocity uncertainties for sources when we ignore TGAS par-allaxes (upper panel) and when we use TGAS parallaxes (lower panel). Ineach case we show the pdfs in vR (black), vz (red), and vϕ (blue). The dashedlines show the median values in each case, which are 6.6 and 3.8 km s−1(without TGAS and with TGAS, respectively) for vR, 4.7 and 2.9 km s−1for vz, and 5.1 and 3.0 km s−1 for vϕ , i.e. the velocity uncertainty in eachdirection is reduced by ∼40 per cent.

8 AGE ESTIMATES

The classical method for determining the age of a star is by com-paring the luminosity of an F or G star to that expected for starsof its colour on the main sequence or turning off it. This is onlypossible if an independent estimate of its distance (e.g. a parallax) isavailable. By including TGAS parallaxes in the RAVE pipeline weare making precisely this comparison, with additional informationand a sophisticated statistical treatment. We can therefore expectthat the ages we derive are as reliable as any currently available formain-sequence stars.

Figure 23. Estimated parallax correction factors (corr� ) for theRAVE+TGAS combined parallax estimates as a function of Teff for dwarfs(log g ≥ 3.5, upper) and log g for giants (log g < 3.5, lower). Values arecalculated in as a running average over a window of width 200 K or 0.3 dex.If we multiply all the RAVE+TGAS � values in this window by corr� ,then � is, on average, zero.

While the original aim of this pipeline was to determine thedistances to stars, an inevitable by-product is that we also constrainthe other ‘model’ properties described in Section 2, i.e. initial massM, age τ , metallicity [M/H], and line-of-sight extinction AV. Wecan also produce new estimates of the other properties of the stars,such as Teff and log g, which we discuss below.

Here, we look at the improved estimates of τ that are madepossible by including TGAS parallaxes. Age estimates from thispipeline were included in RAVE DR4 (in terms of log τ ), but camewith the strong caveat that the prior used (the Standard prior; seeour Section 2.1) included a fixed relationship between metallicityand age (metal-poor stars are assumed to be old, metal-rich starsyounger). In our case, we have now seen that we can use a priorwithout any explicit age–metallicity relationship and still producereasonable results (at least in terms of parallaxes – Fig. 12). Thisgives us some confidence that we will not go too badly wrong usingthis prior when deriving ages.

We would expect that the addition of the TGAS parallax mea-surements provides us with substantial leverage when determiningthe ages of stars, and in Fig. 24 we quantify this. It is clear that,particularly at the low-uncertainty end, we do have a substantialimprovement in precision. Without TGAS only 1.5 per cent of starshave fractional age uncertainties lower than 0.3, while with TGASthis increases to over 25 per cent. In Fig. 25 we show where in the HRdiagram the stars with the smallest age uncertainties are found. Asone would expect, they are primarily found near the main-sequenceturn-off – it is in this region that stars evolve quickly with age, and itis therefore possible to get an age estimate with small uncertaintieseven with imperfect observations.

It is clear from Fig. 17 that there are still some biases in thedistance estimates for dwarfs with 6000 � Teff � 7000 K (i.e. inthe main-sequence turn-off region), though Fig. 23 suggests thatthese are only at the 1 per cent level. It is reasonable to ask whetherthis implies a bias in the age estimates. We cannot know for sure,



Figure 24. The fraction of sources with a given fractional age uncertainty(σ τ /τ ) displayed as a pdf found using a kernel density estimation (lowerpanel), and as a cumulative distribution (upper panel). The median values areplotted as dashed lines. The plot shows the distribution of age uncertaintieswith and without TGAS parallaxes (blue and green curves, respectively).It is particularly clear that the inclusion of TGAS parallaxes allows us toderive age uncertainties of less than 30 per cent for a significant fraction ofsources.

because we do not know what causes the bias. We have investigatedthe possible biases by running the pipeline having either artificiallydecreased the input logg values by 0.4 dex or artificially decreasedthe input parallaxes by 50 μas. Either change results in parallaxestimates that are biased in the opposite sense to that seen with thereal data for these stars. In both cases the changes in stellar agesare small compared to the uncertainties. The change in input log gproduces a typical change of ∼0.5 Gyr (or 5–10 per cent) but withno trend to higher or lower ages (i.e. no clear bias). The change ininput � produces a smaller typical change of ∼0.2 Gyr (or ∼4 percent) with a bias in the sense that cooler stars (with Teff∼ 6000)have slightly lower ages than those originally quoted, by ∼0.1 Gyr(or ∼2 per cent). These are negligible for most purposes, but it isentirely possible that other biases in the analysis (for example in themetallicities or the stellar isochrones) have larger impacts on theage estimates. The study of the complex interplay of these differentfactors is beyond the scope of this paper.

We must caution that these age estimates are extremely hard toverify from external sources. A relatively small number of sourceshave age estimate

improved distances and ages for stars common to tgas ...tgas and dr5 for the sources common to both...

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